See e.g. [1]} Proposition 3.3 for a proof of (REF ). Then the desired result in the non-smooth case follows from a limiting argument.
To see this, fix any \(u\)  that satisfies \(Lu=0\)  in \(\mathbb {R}^{n+1}_+\)  and \(\left\Vert u\right\Vert _{L^\infty (\mathbb {R}^{n+1}_+)}\le 1\) . Then fix a cube \(Q\subset \mathbb {R}^n\)  and define the cutoff function \(\Psi =\Psi _{Q,\epsilon ,\eta }\)  as in (REF ). Take cubes \(\widetilde{Q}_0\)  and \(\widetilde{Q}_1\)  such that
\(\operatorname{supp}\Psi \subset \subset \widetilde{Q}_0\subset \subset \widetilde{Q}_1\subset \subset \mathbb {R}^{n+1}_+.\) 
